Question

There are 14 places in the circle. How many ways can 10 boys and 4 girls sit provided that they do not sit together?

Answer 1

To determine the number of ways 10 boys and 4 girls can sit in 14 places in a circle such that the 4 girls do not sit together, we'll follow these steps:
1. Fix the Circle Arrangement for Boys: Since the arrangement is circular, fixing one boy in place will eliminate identical rotations. Thus, we arrange the remaining 9 boys in \(9!\) ways.
2. Place the Girls: To ensure the girls do not sit together, we consider the gaps created by the 10 boys. In a circle of 10 boys, there are 10 gaps where the girls can be placed.
3. Select 4 Gaps for the Girls: We choose 4 out of these 10 gaps for the girls. This can be done in \(\binom{10}{4}\) ways.
4. Arrange the Girls in the Selected Gaps: The 4 girls can be arranged in the chosen 4 gaps in \(4!\) ways.Thus, the total number of ways to arrange the boys and girls under the given conditions is:
\[ 9! \times \binom{10}{4} \times 4! \]
Now, let's compute the values:
- \(9! = 362,880\)
- \(\binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210\)
- \(4! = 24\)
Multiplying these values together gives us the total number of arrangements:

\[ 9! \times \binom{10}{4} \times 4! = 362,880 \times 210 \times 24 \]

Let's compute this step-by-step:

\[ 210 \times 24 = 5040 \]
\[ 362,880 \times 5040 = 1,830,758,400 \]

Therefore, the number of ways 10 boys and 4 girls can sit in 14 places in a circle such that the girls do not sit together is \(1,830,758,400\).